A General Theory of Sample Complexity for Multi-Item Profit Maximization

One of the most tantalizing and long-standing open problems in mechanism design is profit maximization in multi-item, multi-buyer settings. Much of the literature surrounding this problem rests on the strong assumption that the mechanism designer knows the distribution over buyers' values. In reality, this information is rarely available. The support of the distribution alone is often doubly exponential, so obtaining and storing the distribution is impractical. We relax this assumption and instead assume that the mechanism designer only has a set of samples from the distribution. We develop learning-theoretic foundations of sample-based mechanism design. In particular, we provide generalization guarantees which bound the difference between the empirical profit of a mechanism over a set of samples and its expected profit on the unknown distribution. In this paper, we present a general theory for deriving worst-case generalization bounds in multi-item settings, as well as data-dependent guarantees when the distribution over buyers' values is well-behaved. We analyze mechanism classes that have not yet been studied in the sample-based mechanism design literature and match or improve over the best-known guarantees for many of the special classes that have been studied. The classes we study include randomized mechanisms, pricing mechanisms, multi-part tariffs, and generalized VCG auctions such as affine maximizer auctions.